aureooms-js-integer
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integer code bricks for JavaScript
157 lines (137 loc) • 15.1 kB
JavaScript
;
Object.defineProperty(exports, "__esModule", {
value: true
});
exports.bkaratsuba_t = bkaratsuba_t;
/**
* /!\ BLOCK MULTIPLICATION RESULT MUST HOLD IN THE JAVASCRIPT NUMBER TYPE (DOUBLE i.e. 53 bits)
*
*
* @param {function} add addition algorithm
* @param {function} sub subtraction algorithm
* @param {function} mul multiplication algorithm
* @param {function} copy copy algorithm
* @param {function} calloc array allocator
* @param {uint} r base (radix)
*
*
* EXPLANATION
* ###########
*
* We consider the numbers a and b, both of size N = 2n.
*
* We divide a and b into their lower and upper parts.
*
* a = a1 r^{n} + a0 (1)
* b = b1 r^{n} + b0 (2)
*
* We express the product of a and b using their lower and upper parts.
*
* a b = (a1 r^{n} + a0) (b1 r^{n} + b0) (3)
* = a1 b1 r^{2n} + (a1 b0 + a0 b1) r^{n} + a0 b0 (4)
*
* This gives us 4 multiplications with operands of size n.
* Using a simple trick, we can reduce this computation to 3 multiplications.
*
* We give the 3 terms of (4) the names z0, z1 and z2.
*
* z2 = a1 b1
* z1 = a1 b0 + a0 b1
* z0 = a0 b0
*
* a b = z2 r^{2n} + z1 r^{n} + z0
*
* We then express z1 using z0, z2 and one additional multiplication.
*
* (a1 + a0)(b1 + b0) = a1 b1 + a0 b0 + (a1 b0 + a0 b1)
* = z2 + z0 + z1
*
* z1 = (a1 + a0)(b1 + b0) - z2 - z0
*
* AN ANOTHER WAY AROUND (not used here)
*
* (a1 - a0)(b1 - b0) = (a1 b1 + a0 b0) - (a1 b0 + a0 b1)
* (a0 - a1)(b1 - b0) = (a1 b0 + a0 b1) - (a1 b1 + a0 b0)
* a b = (r^{2n} + r^{n})a1 b1 + r^{n}(a0 - a1)(b1 - b0) + (r^{n} + 1)a0 b0
*
* This algorithm is a generalization of the Toom-Cook algorithm, when m = n = 2.
*
* For further reference, see
* - http://en.wikipedia.org/wiki/Karatsuba_algorithm
* - http://en.wikipedia.org/wiki/Toom–Cook_multiplication
*/
function bkaratsuba_t(add, sub, mul, calloc, mov, r, wrap) {
/**
* Multiply two big endian arrays using karatsuba algorithm,
* i >= j, k >= 2 * i
*
*
* @param {array} a first operand
* @param {int} ai a left
* @param {int} aj a right
* @param {array} b second operand
* @param {int} bi b left
* @param {int} bj b right
* @param {array} c result, must be 0 initialized
* @param {int} ci c left
* @param {int} cj c right
*/
var karatsuba = function karatsuba(a, ai, aj, b, bi, bj, c, ci, cj) {
var z0, z2, t1, t2, t3, n, I, N, N_, i_, j_, i, j, k;
i = aj - ai;
j = bj - bi;
k = cj - ci;
// EMPTY CASE
if (i <= 0 || j <= 0 || k <= 0) return;
// BASE CASE i = j = 1
if (i === 1) {
z0 = a[ai] * b[bi];
c[cj - 1] = z0 % r;
if (k > 1) {
c[cj - 2] = (z0 - c[cj - 1]) / r;
}
}
// RECURSION
else {
n = Math.ceil(i / 2);
I = i + j;
N = 2 * n;
N_ = I - N;
i_ = aj - n;
j_ = Math.max(bi, bj - n);
t1 = calloc(n + 1); // + 1 to handle addition overflows
t2 = calloc(n + 1); // and guarantee reducing k for the
t3 = calloc(N + 1); // recursive calls
z2 = calloc(N_);
z0 = calloc(N);
// RECURSIVE CALLS
mul(a, ai, i_, b, bi, j_, z2, 0, N_); // z2 = a1.b1
mul(a, i_, aj, b, j_, bj, z0, 0, N); // z0 = a0.b0
add(a, i_, aj, a, ai, i_, t1, 0, n + 1); // (a0 + a1)
add(b, bi, j_, b, j_, bj, t2, 0, n + 1); // (b1 + b0)
mul(t1, 1, n + 1, t2, 1, n + 1, t3, 1, N + 1); // (a0 + a1)(b1 + b0)
// BUILD OUTPUT
mov(z2, 0, N_, c, cj - I); // + z2 . r^{2n}
mov(z0, 0, N, c, cj - N); // + z0
if (t1[0]) {
// overflow on t1, add t2 . r^{n}
add(t3, 0, N + 1 - n, t2, 1, n + 1, t3, 0, N + 1 - n);
}
if (t2[0]) {
// overflow on t2, add t1 . r^{n}
add(t3, 0, N + 1 - n, t1, 1, n + 1, t3, 0, N + 1 - n);
}
if (t1[0] && t2[0]) {
// overflow on t1 and t2, add 1 . r^{n+1}
add(t3, 0, N - n, t1, 0, 1, t3, 0, N - n);
}
add(c, ci, cj - n, t3, 0, N + 1, c, ci, cj - n); // + (a0 + a1)(b1 + b0) . r^{n}
sub(c, ci, cj - n, z2, 0, N_, c, ci, cj - n); // - z2 . r^{n}
sub(c, ci, cj - n, z0, 0, N, c, ci, cj - n); // - z1 . r^{n}
}
};
if (wrap !== undefined) karatsuba = wrap(karatsuba);
if (mul === undefined) mul = karatsuba;
return karatsuba;
}
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* /!\\ BLOCK MULTIPLICATION RESULT MUST HOLD IN THE JAVASCRIPT NUMBER TYPE (DOUBLE i.e. 53 bits)\n *\n *\n * @param {function} add addition algorithm\n * @param {function} sub subtraction algorithm\n * @param {function} mul multiplication algorithm\n * @param {function} copy copy algorithm\n * @param {function} calloc array allocator\n * @param {uint} r base (radix)\n *\n *\n * EXPLANATION\n * ###########\n *\n * We consider the numbers a and b, both of size N = 2n.\n *\n * We divide a and b into their lower and upper parts.\n *\n * a = a1 r^{n} + a0 (1)\n * b = b1 r^{n} + b0 (2)\n *\n * We express the product of a and b using their lower and upper parts.\n *\n * a b = (a1 r^{n} + a0) (b1 r^{n} + b0) (3)\n *     = a1 b1 r^{2n} + (a1 b0 + a0 b1) r^{n} + a0 b0 (4)\n *\n * This gives us 4 multiplications with operands of size n.\n * Using a simple trick, we can reduce this computation to 3 multiplications.\n *\n * We give the 3 terms of (4) the names z0, z1 and z2.\n *\n * z2 = a1 b1\n * z1 = a1 b0 + a0 b1\n * z0 = a0 b0\n *\n * a b  = z2 r^{2n} + z1 r^{n} + z0\n *\n * We then express z1 using z0, z2 and one additional multiplication.\n *\n * (a1 + a0)(b1 + b0) = a1 b1 + a0 b0 + (a1 b0 + a0 b1)\n *                    = z2 + z0 + z1\n *\n * z1 = (a1 + a0)(b1 + b0) - z2 - z0\n *\n * AN ANOTHER WAY AROUND (not used here)\n *\n * (a1 - a0)(b1 - b0) = (a1 b1 + a0 b0) - (a1 b0 + a0 b1)\n * (a0 - a1)(b1 - b0) = (a1 b0 + a0 b1) - (a1 b1 + a0 b0)\n * a b = (r^{2n} + r^{n})a1 b1 + r^{n}(a0 - a1)(b1 - b0) + (r^{n} + 1)a0 b0\n *\n * This algorithm is a generalization of the Toom-Cook algorithm, when m = n = 2.\n *\n * For further reference, see\n *  - http://en.wikipedia.org/wiki/Karatsuba_algorithm\n *  - http://en.wikipedia.org/wiki/Toom–Cook_multiplication\n */\n\nexport function bkaratsuba_t (add, sub, mul, calloc, mov, r, wrap){\n\n\t/**\n\t * Multiply two big endian arrays using karatsuba algorithm,\n\t * i >= j, k >= 2 * i\n\t *\n\t *\n\t * @param {array} a first operand\n\t * @param {int} ai a left\n\t * @param {int} aj a right\n\t * @param {array} b second operand\n\t * @param {int} bi b left\n\t * @param {int} bj b right\n\t * @param {array} c result, must be 0 initialized\n\t * @param {int} ci c left\n\t * @param {int} cj c right\n\t */\n\n\tvar karatsuba = function(a, ai, aj, b, bi, bj, c, ci, cj){\n\n\t\tvar z0, z2, t1, t2, t3, n, I, N, N_, i_, j_, i, j, k;\n\n\t\ti = aj - ai;\n\t\tj = bj - bi;\n\t\tk = cj - ci;\n\n\t\t// EMPTY CASE\n\t\tif (i <= 0 || j <= 0 || k <= 0) return;\n\n\t\t// BASE CASE i = j = 1\n\t\tif (i === 1) {\n\n\t\t\tz0 = a[ai] * b[bi];\n\t\t\tc[cj-1] = z0 % r;\n\n\t\t\tif (k > 1) {\n\t\t\t\tc[cj-2] = (z0 - c[cj-1]) / r;\n\t\t\t}\n\n\t\t}\n\n\t\t// RECURSION\n\t\telse{\n\t\t\tn  = Math.ceil(i / 2);\n\t\t\tI  = i + j;\n\t\t\tN  = 2 * n;\n\t\t\tN_ = I - N;\n\t\t\ti_ = aj - n;\n\t\t\tj_ = Math.max(bi, bj - n);\n\n\t\t\tt1 = calloc(n + 1); // + 1 to handle addition overflows\n\t\t\tt2 = calloc(n + 1); // and guarantee reducing k for the\n\t\t\tt3 = calloc(N + 1); // recursive calls\n\t\t\tz2 = calloc(N_);\n\t\t\tz0 = calloc(N);\n\n\t\t// RECURSIVE CALLS\n\t\t\tmul(a, ai, i_, b, bi, j_, z2, 0, N_);            // z2 = a1.b1\n\t\t\tmul(a, i_, aj, b, j_, bj, z0, 0, N);             // z0 = a0.b0\n\t\t\tadd(a, i_, aj, a, ai, i_, t1, 0, n + 1);         // (a0 + a1)\n\t\t\tadd(b, bi, j_, b, j_, bj, t2, 0, n + 1);         // (b1 + b0)\n\t\t\tmul(t1, 1, n + 1, t2, 1, n + 1, t3, 1, N + 1);   // (a0 + a1)(b1 + b0)\n\n\t\t// BUILD OUTPUT\n\t\t\tmov(z2, 0, N_, c, cj - I);                       // + z2 . r^{2n}\n\t\t\tmov(z0, 0, N , c, cj - N);                       // + z0\n\n\t\t\tif (t1[0]) {\n\t\t\t\t// overflow on t1, add t2 . r^{n}\n\t\t\t\tadd(t3, 0, N + 1 - n, t2, 1, n + 1, t3, 0, N + 1 - n);\n\t\t\t}\n\n\t\t\tif (t2[0]) {\n\t\t\t\t// overflow on t2, add t1 . r^{n}\n\t\t\t\tadd(t3, 0, N + 1 - n, t1, 1, n + 1, t3, 0, N + 1 - n);\n\t\t\t}\n\n\t\t\tif (t1[0] && t2[0]) {\n\t\t\t\t// overflow on t1 and t2, add 1 . r^{n+1}\n\t\t\t\tadd(t3, 0, N - n, t1, 0, 1, t3, 0, N - n);\n\t\t\t}\n\n\t\t\tadd(c, ci, cj - n, t3, 0, N + 1, c, ci, cj - n); // + (a0 + a1)(b1 + b0) . r^{n}\n\t\t\tsub(c, ci, cj - n, z2, 0, N_, c, ci, cj - n);    // - z2 . r^{n}\n\t\t\tsub(c, ci, cj - n, z0, 0, N, c, ci, cj - n);     // - z1 . r^{n}\n\t\t}\n\n\t};\n\n\tif (wrap !== undefined) karatsuba = wrap(karatsuba);\n\tif (mul === undefined) mul = karatsuba;\n\n\treturn karatsuba;\n\n}\n"]}